Topology (Math 290, 2016-17)

Justin Roberts

Algebraic topology occupies a very important position in modern mathematics. In many ways it is a microcosm of 20th century mathematics, illustrating features such as the increasing emphasis on global questions, the importance of functoriality, and the use of rather general and abstract machinery to solve quite specific problems. Since many of these developments (for example homological algebra, which is very important in subjects such as algebraic geometry and number theory) actually have their roots in algebraic topology, it can be very useful even for those not directly interested in its subject matter to have some familiarity with its methods, especially as it is probably the most accessible of the subjects which require "heavy machinery".

The heart of algebraic topology is the study of properties of arbitrary topological spaces, considered up to homotopy equivalence. This is a very general class of spaces, and the equivalence is a fairly brutal kind of deformation - but one can use it to study finer structures too. In fact it has enormous numbers of applications in areas such as low-dimensional topology (topics such as knot theory and three-manifold theory); the classification of high-dimensional manifolds up to homeomorphism; the study of vector fields on manifolds (and other hard questions of linear algebra); fixed point theory; the study of spaces of functions and of solutions of differential equations (index theory); combinatorics; and all manner of uses in modern geometry and physics.

I'm a low-dimensional topologist by training, and I generally favour geometric over algebraic examples, but I also find the category-theoretic view of the world extremely useful, and will try to exhibit connections with other areas of mathematics where possible (since algebraic topology is much more of a "service" course these days than it used to be).

Here's a rough outline of what I expect to cover:

290A: The fundamental group, category theory, van Kampen's theorem, CW-complexes, covering spaces, simplicial and singular homology.

290B: Homology theory, cellular homology, cohomology, homological algebra, manifolds, Poincare duality and intersection theory. (Other possibilities: spectral sequences, sheaves and Cech cohomology.)

290C Basic homotopy theory, homotopy groups, cellular methods, Hurewicz and Whitehead theorems. (This is as far as qualifying exam material usually goes, though if there's time I'll also sketch a few additional topics from the lists below.)

Imaginary subsequent courses:

290D: Further algebraic topology: fibrations, fibre and vector bundles, classifying spaces, Eilenberg-MacLane spaces, obstruction theory, characteristic classes, Postnikov towers, simplicial sets, cohomology theories and spectra, K-theory, Bott periodicity.

290E: Geometric topology: differential topology, Morse theory, handle theory, the h-cobordism theorem and high-dimensional smooth Poincare conjecture, 3-manifolds, 4-manifolds.

290F Modern algebraic topology: differential graded algebras, rational homotopy theory, derived categories, model categories, infinity algebras, homotopical algebra, towards derived/homotopical geometry.

Lectures. MWF 10-10.50 in AP&M 5402.

Office hours.
Will be ??????????  in room 7210, AP&M, or by appointment. My email is and phone number is 534-2649.

TA. The TA is Jack Geller: in AP&M 6452. His section will be in 5829 on Mondays from 2-3(ish)pm; his office hours are ????????????? (or by appointment).

Homework. Homework problems are assigned primarily for educational purposes rather than for assessment, so they are of varying difficulty and aren't necessarily going to be of "qualifying exam standard". They're not going to be officially graded (the whole class grade will be based on the final exam, which will be as close to qualifying exam standard as I can make it) but you should at least think about them for discussion with Jack, and preferably try to write down solutions to as many as you can so that he can comment on them!

Now writing is definitely not the best way of communicating mathematics! It's much more effective to have a conversation, where you can experiment with "handwavy" ideas and dynamically add or gloss over details according to what your audience asks for or about. Writing, because it is static, gives you just one shot at communicating your ideas (perhaps even to later generations). Particularly in topology, it can be hard to strike the right balance between "putting in all the necessary details" and "boring the reader with unnecessary details". Obviously one has to err on the side of comprehensibility, so my recommendation is that when writing maths you should usually imagine that you're writing to a friend with a similar background to your own, but who is slightly less clever than you are! This friend can be expected to fill in genuinely routine "boring" details but will expect you to mention any standard facts or theorems you need to use. For homework therefore, just imagine you are writing a solution for a confused classmate...

(Below you can see the old qualifying exams which I've set; you should be able to see that they are much more consistent in style than the HW!)

Problem sheet 1

Problem sheet 2

Problem sheet 3

Problem sheet 4

Problem sheet 6

Problem sheet 7

Problem sheet 8

Problem sheet 9

Problem sheet 10

Problem sheet 11

Problem sheet 12

Problem sheet 13

Problem sheet 14

Old quals

Since most students already know it (and it is being revisited in the analysis qual courses), I won't talk about point-set topology at all. I have some rather rough notes on point-set topology which might serve as a reminder. The algebraic background required is quite elementary: knowing what groups, rings, fields, bilinear forms and dual spaces are should suffice, as I'll try to explain any more sophisticated structures myself.

Books. I don't work from a book (either for lecturing or setting problems), but Algebraic Topology by Allen Hatcher (Cambridge University Press) is the now-standard book on this material. It is available for free downloading from his webpage, but it's probably less hassle to simply buy it, as it's cheap!

Other useful recent books on algebraic topology are Peter May's
A concise course in algebraic topology and Dodson and Parker's User's guide to algebraic topology. These both have a sort of guidebook flavour; they try to cover a lot of ground more quickly than Hatcher does, without swamping the reader with detail. There are plenty of older textbooks too: Greenberg and Harper is one of the best. For serious reference there are books such as Spanier, Switzer, and George Whitehead's Homotopy theory, but these are much less readable. For the very basics of point-set and algebraic topology the best book is Armstrong's Basic Topology (Springer).

A quite different sort of book is Glen Bredon's
Topology and Geometry (Springer), which deals with differential topology as well as algebraic topology. I like the fact that it tries to treat algebraic topology more in the context of geometry on manifolds (and ideally I would like to teach a topology course which mixed algebraic topology with differential topology all along), but students sometimes find it a bit disorganised. Perhaps for this approach it would be better to turn to the classic references: Milnor's Topology from the differentiable viewpoint (Princeton) and Bott and Tu's Differential froms in algebraic topology (Springer).

(A picture of the Hopf fibration from Rob Scharein's KnotPlot Site.)